Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,20,15)(16,18,22,19)(17,21,25,23)(24,26,28,27);(1,2,3,4)(7,8)(10,12)(13,15,18,22,20,14,19,16)(17,23,26,28,25,21,27,24);(5,7,9,8)(6,10,11,12)(13,16,19,14,20,22,18,15)(17,24,27,21,25,28,26,23);(5,8,9,7)(6,10,11,12)(13,17,20,25)(14,21,15,23)(16,24,22,28)(18,26,19,27);(6,11)(10,12)(17,25)(21,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390661);